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Dual Galilean invariance in a temporally dispersive medium

Ioannis M. Besieris and Peeter Saari

Open Access Open Access

Abstract

The bidispersive equation, accounting for second-order temporal dispersion in a lossless medium, is examined under a Galilei-like transformation, as well as a conditional ordinary Galilei transformation. In the former case, the roles of space and time are reversed by comparison to the application of the same transformation to the quantum mechanical Schrödinger equation. Such an invariance can result in envelope speeds that can assume values above or below the group speed of the medium; also, they can be negative. Conditional Galilei transformation results in a decrease in dimensionality.

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1. Introduction

It is well known that the Galilean transformation $z^{\prime} = z - vt,\;t^{\prime} = t$ arises from the “subluminal” Lorentz transformation

$$z^{\prime} = {\gamma _{sub}}({z - vt} ),\;x^{\prime} = x,\;y^{\prime} = y,\;t^{\prime} = {\gamma _{sub}}({t - zv/{c^2}} );\;{\gamma _{sub}} = 1/\sqrt {1 - {{(v/c)}^2}} ;\;v < c$$
in the limit $|v |/c \to 0.$ Consider, specifically, the 3D Schrödinger equation
$$ih\frac{\partial }{{\partial t}}\varphi ({x,y,z,t} )+ \frac{{{h^2}}}{{2m}}\left( {\nabla_t^2 + \frac{{{\partial^2}}}{{\partial \,{z^2}}}} \right)\varphi ({x,y,z,t} )= 0.$$

The Galilean transformation changes Eq. (2) as follows:

$$ih\left( { - v\frac{\partial }{{\partial z^{\prime}}} + \frac{\partial }{{\partial t^{\prime}}}} \right)\varphi ({x,y,z^{\prime},t^{\prime}} )+ \frac{{{h^2}}}{{2m}}\left( {\nabla_t^2 + \frac{{{\partial^2}}}{{\partial \,{{z^{\prime}}^2}}}} \right)\varphi ({x,y,z^{\prime},t^{\prime}} )= 0.$$

Obviously, the free particle Schrödinger equation is not form-invariant under this transformation. The first term on the left-hand side requires an extra phase to be added to the wavefunction in order to achieve form-invariance. This problem can be resolved using an extension of a technique introduced by Kennard [1] and Darwin [2] in the early days of quantum mechanics. Specifically, given a solution to Eq. (2), the expression

$$\psi ({x,y,z,t} )= \exp \left[ {i\frac{{mv}}{h}\left( {z - \frac{1}{2}vt} \right)} \right]\varphi ({x,y,z - vt,t} )$$
is also a solution.

Choosing a finite-energy solution $\varphi ({x,y,z,t} )$ to Eq. (2), replacing in it $z$ by $z - vt$ and using the resulting expression in Eq. (4) will result in a finite-energy, localized, and linearly traveling wave solution to the 3D Schrödinger equation. The “envelope function” $\varphi ({x,y,z - v\,t,t} )$ travels with the speed $v$. However, the phase term in Eq. (4) travels at the speed $v/2$. The explicit dependence of $\varphi ({x,y,z - v\,t,t} )$ on time signifies that the wave function $\psi (x,y,z,t)$ will eventually disperse.

To derive an illustrative example of a finite-energy wave function $\psi (x,y,z,t)$, consider the following specific axisymmetric solution to Eq. (2) in cylindrical coordinates $({\rho ,\phi ,z})$:

$$\varphi (\rho ,z,t) = \frac{{\sqrt \pi }}{4}{\left( {\frac{{2m\hbar }}{{{t_0} + it}}} \right)^{3/2}}\exp \left[ { - \frac{m}{{2\hbar }}\left( {\frac{{{\rho^2} + {z^2}}}{{{t_0} + it}}} \right)} \right].$$

Here, ${t_0}$ is an arbitrary positive parameter with units of time. Using this expression in conjunction with Eq. (4), one obtains the finite-energy traveling wave wavefunction

$$\psi ({\rho ,z,t} )= \exp \left[ {\frac{i}{h}mv\left( {z - \frac{1}{2}vt} \right)} \right]\frac{{\sqrt \pi }}{4}{\left( {\frac{{2m\hbar }}{{{t_0} + it}}} \right)^{3/2}}\exp \left[ { - \frac{m}{{2\hbar }}\left( {\frac{{{\rho^2} + {{({z - vt} )}^2}}}{{{t_0} + it}}} \right)} \right].$$

This specific solution was first derived by Darwin [1]. As the envelope moves along the z- direction with speed v, it spreads out (both along $\rho $ and z) and its amplitude decreases. However, for large values of the parameter ${t_0}$ (essentially for a long pulse), the solution remains localized up to a time t of $O({{t_0}} ).$

A variant of the Galilean transformation, specifically $x^{\prime} = x - az,\;z^{\prime} = z,$ has been useful in connection with the parabolic equation

$$ik\frac{\partial }{{\partial z}}\varphi ({x,z} )+ \frac{1}{2}\frac{{{\partial ^2}}}{{\partial \,{x^2}}}\varphi ({x,z,} )= 0$$
governing paraxial monochromatic wave propagation in free space. Here, $k = {\omega _0}/c$ denotes a wavenumber, defined in terms of a carrier frequency and the speed of light in vacuum. Used, for example, in connection with an “accelerating” Airy beam solution to Eq. (7), it results in ballistic-like properties of the resulting beam [8].

Although both the Schrodinger and the paraxial equations are not form-invariant under a Galilean transformation, this is not the case for other types of equations. It has been found that form invariance under a Galilean transformation is feasible for certain nonlinear equations governing wave propagation in temporally dispersive media. Among them are the Korteweg-de Vries and the Benjamin-Bona-Mahony equations [9].

Consider, next, the “superluminal” Lorentz transformation

$$z^{\prime} = {\gamma _{\sup }}({z - t{c^2}/v} ),\;x^{\prime} = x,\;y^{\prime} = y,\;t^{\prime} = {\gamma _{\sup }}({t - z/v} );\;{\gamma _{\sup }} = 1/\sqrt {1 - {{(c/v)}^2}}$$
for $v > c.$ In the limit $|v |/c \to \infty ,$ one obtains the dual Galilean transformation $t^{\prime} = t - z/v,\;z^{\prime} = z.$ More generally, the group of space-time transformations, $\vec{r}^{\prime} = \vec{r},\;t^{\prime} = t - \vec{a} \cdot \vec{r},$ with the roles of space and time interchanged in comparison to the regular Galilean transformation, was studied by Lévy-Leblond [10] and Sen Gupta [11].

It is our goal in this article to study the implications of the dual Galilean transformation to the bidispersive equation describing paraxial electromagnetic wave propagation in a lossless temporally dispersive medium. A derivation of this equation, the application of the dual Galilean transformation for both normal and anomalous dispersion, and illustrative examples are given in Sec. 2. Two Ansätzen that facilitate solutions to the bidispersive equation for both normal and anomalous dispersion are discussed and illustrated with specific examples in Sec. 3. It is shown in Sec. 4 that the bidispersive equation can be conditionally invariant under an ordinary Galilean transformation. However, this entails a decrease in dimensionality. Concluding remarks are provided in Sec. 5.

2. Dual Galilean transformation of the bidispersive equation in a temporally dispersive medium

2.1 Bidispersive equation

Electromagnetic wave propagation in a linear, homogeneous, transparent, dispersive medium is governed by the scalar equation

$${\nabla ^2}u(\vec{r},t) + \beta _{op}^2( - i\partial /\partial t)u(\vec{r},t) = 0.$$

In this expression, $u(\vec{r},t)$ is a real scalar field (a component of the vector-valued electric or magnetic fields) and $\beta _{op}^2( - i\partial /\partial t) = {({\omega /c} )^2}{n^2}( - i\partial /\partial t),n( - i\partial /\partial t)$ being the index of refraction, is a real pseudo-differential operator. A physical interpretation of the latter is provided in the frequency domain; specifically,

$$F\{{\beta_{op}^2( - i\partial /\partial t)\,u(\vec{r},t)} \}= {\beta ^2}(\omega ){\kern 1pt} \,\tilde{u}(\vec{r},\omega )$$
where $F\{{\cdot} \}$ denotes Fourier transformation and $\tilde{u}(\vec{r},\omega )$ is the Fourier transform of $u(\vec{r},t)$ with respect to time. The function $\beta (\omega )$ appearing on the right-hand side of Eq. (10) is a real wave number for a transparent dispersive medium.

For a physically convenient central radian carrier frequency ${\omega _0}$, the real field $u(\vec{r},t)$ is expressed as follows:

$$u(\vec{r},t) = \varphi (\vec{r},t)\,exp [{ - i{\omega_0}({t - z/{v_{ph}}} )} ]\, + cc,\;\;z \ge 0.$$

Here, $\varphi (\vec{r},t)$ is a complex-valued envelope function and ${v_{ph}} = {\omega _0}/\beta ({\omega _0})$ denotes the phase speed in the medium computed at the central frequency ${\omega _0}$. A formal introduction of Eq. (11) into Eq. (9) yields the following exact equation governing the envelope function $\varphi (\vec{r},t)$[3–7]:

$$\left[ {\nabla_ \bot^2 + \frac{{{\partial^2}}}{{\partial {z^2}}} + 2i{\beta_c}({\omega_0})\frac{\partial }{{\partial z}} - {\beta^2}({\omega_0}) + \sum\limits_{m = 0}^\infty {\frac{1}{{m!}}\frac{{{\partial^m}}}{{\partial {\omega^m}}}{\beta^2}(\omega )\left|\begin{array}{l} \\ \omega = {\omega_0} \end{array} \right.{{\left( {i\frac{\partial }{{\partial t}}} \right)}^m}} } \right]\varphi (\vec{r},t) = 0.$$

Here, $\nabla _ \bot ^2$ denotes the transverse (with respect to $z$) Laplacian operator. Usually, at this stage in the study of wave propagation through dispersive media, one introduces the moving reference frame $\xi = z,\;\;\tau = t - (z/{v_{gr}})$, in terms of the group speed ${v_{gr}} = 1/{\beta _1};\;\;{\beta _1} \equiv { {d\beta (\omega )/d\omega } |_{\omega = {\omega _0}}}$. Then, Eq. (12) is transformed into the expression

$$\begin{array}{l} \left[ {\nabla_ \bot^2 + \frac{{{\partial^2}}}{{\partial {z^2}}} + \frac{1}{{v_{gr}^2}}\frac{{{\partial^2}}}{{\partial {\tau^2}}} - 2\frac{1}{{{v_{gr}}}}\frac{{{\partial^2}}}{{\partial z\partial \tau }} + 2i\beta ({\omega_0})\left( {\frac{\partial }{{\partial z}} - \frac{1}{{{v_{gr}}}}\frac{\partial }{{\partial \tau }}} \right) - {\beta^2}({\omega_0})} \right]\varphi (\vec{r},\tau )\\ \\ + \sum\limits_{m = 0}^\infty {\frac{1}{{m!}}\frac{{{\partial ^m}}}{{\partial {\omega ^m}}}{\beta ^2}(\omega )\left|\begin{array}{l} \\ \omega = {\omega_0} \end{array} \right.{{\left( {i\frac{\partial }{{\partial \tau }}} \right)}^m}} \varphi (\vec{r},\tau ) = 0. \end{array}$$

Several techniques have been developed based on the type of approximations made to the exact Eq. (13). In the sequel, use will be made of the slowly varying envelope approximation (SVEA) (cf. e.g., [4]), whereby one neglects the second derivative with respect to z (paraxial approximation), as well as the mixed derivative term involving $z$ and $\tau $. Furthermore, dispersive effects up to the second order will be retained. One, then, has the bidispersive equation

$$\left( {i\frac{\partial }{{\partial z}} \mp \frac{1}{2}{{\bar{\beta }}_2}\frac{{{\partial^2}}}{{\partial {\tau^2}}} + \frac{1}{{2{\beta_0}}}\nabla_ \bot^2} \right){\varphi _ \mp }(x,y,\tau ,z) = 0,\,$$
where ${\bar{\beta }_2} = |{{\beta_2}} |,$ with $\;\;{\beta _2} \equiv { {{d^2}\beta (\omega )/d{\omega^2}} |_{\omega = {\omega _0}}}$ the second-order index of dispersion. The negative sign in Eq. (14) corresponds to normal dispersion and the positive sign to anomalous dispersion. In the latter case, Eq. (14) is homeomorphic to the 3D Schrödinger equation.

2.2 Dual Galilean transformation of the bidispersive equation

Motivated by the dual Galilean transformation $t^{\prime} = t - z/v,\;z^{\prime} = z\;,$ the change of variables $\tau ^{\prime} = \tau - z/v,\;z^{\prime} = z\;\;$ will be applied to the bidispersive Eq. (14) without a restriction on the speed v. The bidispersive equation is not form-invariant under this transformation. A phase term must be added to the wavefunction for this requirement to be fulfilled. Specifically, given a solution to Eq. (14), the expression

$${\psi _ \mp }({x,y,\tau ,z} )= \exp \left[ { \mp i\frac{1}{{v{{\bar{\beta }}_2}}}\left( {\tau - \frac{z}{{2v}}} \right)} \right]{\varphi _ \mp }\left( {x,y,\tau - \frac{z}{v},z} \right)$$
is also a solution, where v is an arbitrary parameter with units of $m/s.$ There is a difference between Eq. (2) for the Schrödinger equation and Eq. (14) for the bidispersive equation. Two consecutive applications of the Galilei transformation in the former case result in an effective envelope speed ${v_{en}} = {v_1} + {v_2}.$ On the other hand, two consecutive applications of the Galilei-like transformation in the latter give rise to the effective envelope speed ${v_1}{v_2}/({{v_1} + {v_2}} ).$ This is in contrast to the following known result: In a medium that is at rest and its oscillations are described by the dispersion relationship $\omega = \omega (k )$ a wavepacket with frequency near the carrier frequency ${\omega _0} = \omega ({{k_0}} )$ has a group speed given by ${v_g} = d\omega ({{k_0}} )/d{k_0}.$ A Galilei transformation in this case results in the effective group speed ${v_{eff}} = {v_g} - v.$

Recall that in Eq. (14) $\tau = t - z/{v_{gr}}.$ It is clear, then, as mentioned above, that the envelope wavefunction no longer is seen to travel at the medium group velocity ${v_g}.$ Instead, one obtains $\tau - \frac{z}{v} = T = t - \frac{z}{{{v_{en}}}},$ with ${v_{en}} = v\,{v_g}{({v + {v_g}} )^{ - 1}}$ the new envelope speed. For $v > 0,$ one has ${v_{en}} < {v_g}.$ In the limiting case $v \to \infty ,$ one obtains ${v_{en}} \to {v_g}.$ Finally, ${v_{en}} > {v_g}$ for $v < - {v_g},$ but the wavepacket moves backwards.

In addition to the new envelope speed, a new phase speed is introduced. From Eq. (15), one obtains $\left( {\tau - \frac{z}{{2v}}} \right) = t - \frac{z}{{{V_{ph}}}},$ where ${V_{ph}} = 2v\,{v_g}{({{v_g} + 2v} )^{ - 1}}$ is the new phase speed. As in the case of the envelope speed, ${V_{ph}}$ is smaller from ${v_g}$ for $v > 0$. A combination of the phase factors in Eqs. (11) and (15), gives rise to an overall effective phase speed $V_{ph}^{eff} = {v_{ph}}\left( {{\beta_0} + \frac{1}{{{{\bar{\beta }}_2}v{v_{ph}}}}} \right){\left( {{\beta_0} + \frac{1}{{{{\bar{\beta }}_2}v{V_{ph}}}}} \right)^{ - 1}}.$

2.3 Illustrative example: normal dispersion

Consider the (3 + 1) D bidispersive equation

$$\left( {i\frac{\partial }{{\partial z}} - \frac{1}{2}{{\bar{\beta }}_2}\frac{{{\partial^2}}}{{\partial {\tau^2}}} + \frac{1}{{2{\beta_0}}}\nabla_t^2} \right)\varphi (\vec{r},\tau ) = 0$$
for normal dispersion. A well-known solution to this problem is the azimuthally symmetric accelerating Airy wavepacket [12,13]
$$\begin{array}{l} \varphi ({\rho ,z,\tau } )= \exp \left[ { - \frac{1}{{12}}({2a - iz} )\left( {2{a^2} - 4iaz + {z^2} - \frac{{6\tau }}{{\sqrt {{{\bar{\beta }}_2}} }}} \right)} \right]Ai\left( {\frac{\tau }{{\sqrt {{{\bar{\beta }}_2}} }} - iaz - \frac{{{z^2}}}{4}} \right)\\ \textrm{ } \times \frac{1}{{{a_1} + iz}}\exp \left[ { - \frac{{{\beta_0}}}{2}\frac{{{\rho^2}}}{{{a_1} + iz}}} \right]. \end{array}$$

Here, ${a_1}\;\textrm{and }a$ are positive parameters, the latter small enough to guarantee the square integrability of the solution. Next, the dual Galilean transformation in Eq. (17) is carried out. Specifically,

$$\psi ({\rho ,\tau ,z} )= \exp \left[ { - i\frac{1}{{v{{\bar{\beta }}_2}}}\left( {\tau - \frac{z}{{2v}}} \right)} \right]\varphi \left( {\rho ,\tau - \frac{z}{v},z} \right)$$

Plots of $|{\varphi ({0,z,\tau } )} |$ and $|{\psi ({0,z,\tau } )} |vs.\textrm{ }\tau \textrm{ and }z$ are shown in Figure 1.

 figure: Fig. 1.

Fig. 1. (a) $|{\varphi ({0,z,\tau } )} |\textrm{ }vs.\textrm{ }\tau \textrm{ and }z;$ with $a = 5 \times {10^{ - 2}}, {a_1} = 10,$ normalized values ${\beta _0} = 2, {\bar{\beta }_2} = 1$ and $v ={-} 1/2,$ (b) $|{\psi ({0,z,\tau } )} |\textrm{ }vs.\textrm{ }\tau \textrm{ and }z,$ with $a = 5 \times {10^{ - 2}},\;{a_1} = 10,\;{\beta _0} = 2,\;{\bar{\beta }_2} = {10^{ - 1}},\;v ={-} 2$.

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2.4 Illustrative example: anomalous dispersion

A specific solution of Eq. (14) in the case of anomalous dispersion is given by

$$\varphi ({\rho ,\tau ,z} )= \frac{1}{{{{({a + iz} )}^{3/2}}}}\exp \left[ { - \frac{1}{2}\frac{{{\beta_0}{\rho^2} + {\tau^2}/{{\bar{\beta }}_2}}}{{a + iz}}} \right],\;a > 0,$$
which transforms to
$$\psi ({\rho ,\tau ,z} )= \frac{1}{{{{({a + iz} )}^{3/2}}}}\exp \left[ {i\frac{1}{{v{{\bar{\beta }}_2}}}\left( {T + \frac{z}{{2v}}} \right)} \right]\exp \left[ { - \frac{1}{2}\frac{{{\beta_0}{\rho^2} + {T^2}/{{\bar{\beta }}_2}}}{{a + iz}}} \right]$$
under the Galilei-like transformation in Eq. (15). Here, $T = t - z/{v_{en}}.$ Fig. 2 shows surface plots of ${\textrm{Re}} \{{\psi (\rho ,T,z} \}\textrm{ vs}\textrm{. }T \textrm{ and }\rho $ at three range positions. Also, temporal plots ${\textrm{Re}} \{{\psi (0,T,z} \}\textrm{ vs}\textrm{. }T .\textrm{ }$

 figure: Fig. 2.

Fig. 2. (a) Surface plot of $|{{\textrm{Re}} \{{\psi (\rho ,T,z} \}} |\textrm{ vs}\textrm{. }T \textrm{ and }\rho $ at three range positions $({z = \,\, - 1/8,\;0,\textrm{ and }1/8} )$; (b) ${\textrm{Re}} \{{\psi (0,T,z} \}\textrm{ vs}\textrm{. }T \textrm{ }$ at three range positions $({z = \,\,0,\;1/4,\textrm{ and }1/2} )$. Parameter values: $a = {10^{ - 1}}$ and normalized values ${\beta _0} = 5,\bar{\beta } = 5 \times {10^{ - 2}}$ and $v = 1.2.$

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3. Two ansätzen for the bidispersive equation

It will be convenient for the following work to introduce the dimensionless variables

$$Z = z/({{\beta_0}x_o^2} ),\;T = \tau /{\tau _0},\;X = x/{x_0},\;Y = y/{x_0}.$$

In this manner, the diffraction length ${L_{diff}} = {\beta _0}x_o^2$ is equal to the corresponding dispersion length ${L_{disp}} = {\tau ^2}/{\bar{\beta }_2}.$ Here, ${x_0}$ is an arbitrary length scale and ${\tau _0} = {x_0}\sqrt {{\beta _0}{{\bar{\beta }}_2}} $ is related to the pulse width of the wavepacket. We have, then, in the place of Eq. (14)

$$\left( {i\frac{\partial }{{\partial Z}} \mp \frac{1}{2}\frac{{{\partial^2}}}{{\partial {T^2}}} + \frac{1}{2}\nabla_ \bot^2} \right)\varphi (X,Y,T,Z) = 0.$$

3.1 Ansatz I. Normal dispersion

Consider any solution of the Klein-Gordon equation

$$\left( {\nabla_t^2 - \frac{{{\partial^2}}}{{\partial {W^2}}} - {\lambda^2}} \right)s({X,Y,W;\lambda } );\;\lambda = {({2 + {a^2} + 2b} )^{1/2}}.$$

Then,

$$\varphi ({X,Y,T,Z} )= {e^{ibZ}}{e^{ - iaT}}{e^{i({1 + {a^2}} )Z}}s[{X,Y,T - aZ;{{({2 + {a^2} + 2b} )}^{1/2}}} ]$$
is a solution to Eq. (22) for normal dispersion. To prove this ansatz, we consider a solution of Eq. (22) with the negative sign of the form
$$\varphi ({X,Y,T,Z} )= \exp ({iqZ} )\Phi \left( {X,Y,T;\sqrt {2q} } \right)$$
and apply the Galilei-like transformation. As a result, we obtain
$$\varphi ({X,Y,T,Z} )= {e^{iqZ}}{e^{ - ia({T - aZ/2} )}}\Phi \left( {X,Y,T - aZ;\sqrt {2q} } \right).$$

With $q = 1 + {a^2}/2 + b,$ we obtain the ansatz given in Eq. (24).

3.1.1 Illustrative example

A nonsingular azimuthally symmetric solution to the Klein-Gordon equation in (23) is given by

$$s({R,W;\lambda } )= \exp \left( { - \lambda \sqrt {{R^2} + {{({{a_1} + iW} )}^2}} } \right)/\sqrt {{R^2} + {{({{a_1} + iW} )}^2}} ,$$
where ${a_1}$ is a positive parameter and $R = {({{X^2} + {Y^2}} )^{1/2}}$ is the polar coordinate. Application of the ansatz in Eq. (24) yields the solution
$$\varphi ({R,T,Z} )= {e^{ibZ}}{e^{ - iaT}}{e^{i({1 + {a^2}} )Z}}\frac{{\exp \left[ { - {{({2 + {a^2} + 2b} )}^{1/2}}\sqrt {{R^2} + {{({{a_1} + i({T - aZ} )} )}^2}} } \right]}}{{\sqrt {{R^2} + {{({{a_1} + i({T - aZ} )} )}^2}} }}.$$

This is an infinite-energy envelope-invariant solution propagating at the dimensionless speed $Z/T = V = 1/a.$ A finite-energy solution can be derived by making use of the free parameter $b.$ One such solution in terms of the original variables and with $a = ({{x_0}/v} ){({{\beta_0}/{{\bar{\beta }}_2}} )^{1/2}}$ is given as follows:

$$\begin{array}{l} \varphi ({\rho ,z,t} )= {e^{iA({z - {v_{ph}}t} )}}\frac{C}{{\sqrt B }}erfc\left( {\frac{{C\sqrt B }}{{\sqrt 2 }}} \right);\\ A = \frac{{v + {v_g}}}{{2{v^2}{v_g}{{\bar{\beta }}_2}}},\;C = {\left( {{a_1} - i\frac{z}{{{\beta_0}x_0^2}}} \right)^{ - 1/2}};\\ B = \frac{{{\rho ^2}}}{{x_0^2}} + {\left[ {{a_1} - i\frac{1}{{{x_0}\sqrt {{\beta_0}{{\bar{\beta }}_2}} }}\left( {\frac{{v + {v_g}}}{{v{v_g}}}} \right)({z - {v_{en}}t} )} \right]^2};\\ \textrm{ }{v_{en}} = \frac{{v{v_g}}}{{v + {v_g}}}\textrm{, }{v_{ph}} = \frac{{2v{v_g}}}{{2v + {v_g}}};\;\rho = \sqrt {{x^2} + {y^2}} . \end{array}$$

Here, $erfc({\cdot} )$ denotes the complementary error function. The envelope and phase speeds are those specified earlier. Fig. 3 shows surface plots of $|{\psi (\rho ,\varsigma } |\textrm{ vs}\textrm{. }\varsigma = z - {v_{en}}\textrm{t and }\rho $ and ${\textrm{Re}} \{{\psi ({\rho ,\varsigma } )} \}\textrm{ vs}\textrm{. }\varsigma $ for $\rho = 0$.

 figure: Fig. 3.

Fig. 3. (a) Surface plot of $|{\psi (\rho ,\varsigma } |\textrm{ vs}\textrm{. }\varsigma = z - {v_{en}}\textrm{t and }\rho $ at three range positions $vt = {v_{en}}t ={-} 30,\,\;0,\textrm{and 30}$ and (b) ${\textrm{Re}} \{{\psi ({\rho ,\varsigma } )} \}\textrm{ vs}\textrm{. }\varsigma $ for $\rho = 0$ at three range positions $ vt = {v_{en}}t = 0,\,\;30,\textrm{ and 50}\textrm{.}$ Dimensionless parameter values: $a{}_1 = 1,\;\;{x_0} = 1$ and normalized values $c = 1,\;v = 4,\;{\beta _0} = 5,\;{\beta _1} = 2,\textrm{ and }{\bar{\beta }_2} = {10^{ - 1}}.$

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3.2 Ansatz II. Anomalous dispersion

Consider a solution of the Helmholtz equation

$$\left( {\nabla_t^2 + \frac{{{\partial^2}}}{{\partial {W^2}}} + {\lambda^2}} \right)s({X,Y,W;\lambda } );\;\lambda = {({2 + {a^2} + 2b} )^{1/2}}.$$

Then,

$$\varphi ({X,Y,T,Z} )= {e^{ - ibZ}}{e^{iaT}}{e^{ - i({1 + {a^2}} )Z}}s[{X,Y,T - aZ;{{({2 + {a^2} + 2b} )}^{1/2}}} ]$$
satisfies Eq. (22) for anomalous dispersion. The proof is analogous to that for normal dispersion.

3.2.1 Illustrative example

A nonsingular azimuthally symmetric solution to the Helmholtz in (30) is given by

$$s({R,W;\lambda } )= \frac{{\sin \left( {\lambda \sqrt {{R^2} + {W^2}} } \right)}}{{\sqrt {{R^2} + {W^2}} }}.$$

Application of the ansatz in Eq. (31) yields the solution

$$\varphi ({R,T,Z} )= {e^{ - ibZ}}{e^{iaT}}{e^{ - i({1 + {a^2}} )Z}}\frac{{\sin \left[ {{{({2 + {a^2} + 2b} )}^{1/2}}\sqrt {{R^2} + {{({T - aZ} )}^2}} } \right]}}{{\sqrt {{R^2} + {{({T - aZ} )}^2}} }}.$$

This is an infinite-energy envelope invariant solution propagating at the dimensionless speed $V = 1/a.$ A finite-energy solution can be derived by making use of the free parameter $b.$ One such solution in terms of the original variables and with $a = ({{x_0}/v} ){({{\beta_0}/{{\bar{\beta }}_2}} )^{1/2}}$ is given as follows:

$$\begin{array}{l} \varphi ({\rho ,z,t} )= {e^{iA({z - {v_{ph}}t} )}}{C^3}{e^{ - {C^2}B/2}};\\ A = \frac{{2v + {v_g}}}{{2{v^2}{v_g}{{\bar{\beta }}_2}}},\;C = {\left( {{a_1} + i\frac{z}{{{\beta_0}x_0^2}}} \right)^{ - 1/2}};\\ B = \frac{{{\rho ^2}}}{{x_0^2}} + {\left[ {\frac{1}{{{x_0}\sqrt {{\beta_0}{{\bar{\beta }}_2}} }}\left( {\frac{{v + {v_g}}}{{v{v_g}}}} \right)({z - {v_{en}}t} )} \right]^2};\\ \textrm{ }{v_{en}} = \frac{{v{v_g}}}{{v + {v_g}}}\textrm{, }{v_{ph}} = \frac{{2v{v_g}}}{{2v + {v_g}}};\;\rho = \sqrt {{x^2} + {y^2}} . \end{array}$$

The envelope and phase speeds are those specified earlier. Fig. 4 shows $|{{\textrm{Re}} \{{\psi ({\rho ,\varsigma } )} \}} |\textrm{ vs}\textrm{. }\varsigma = z - {v_{en}}\textrm{t and }\rho $ and ${\textrm{Re}} \{{\psi ({\rho ,\varsigma } )} \}\textrm{ vs}\textrm{. }\varsigma $ for $\rho = 0$.

 figure: Fig. 4.

Fig. 4. (a) Surface plot of $\left| {{\textrm{Re}} \{{\psi ({\rho ,\varsigma } )} \}} \right|\textrm{ vs}\textrm{. }\varsigma = z - {v_{en}}\textrm{t and }\rho $ at three range positions $vt = {v_{en}}t = - 10,\,\;0,\textrm{and 10}$ and (b) ${\textrm{Re}} \{{\psi ({\rho ,\varsigma } )} \}\textrm{ vs}\textrm{. }\varsigma $ for $\rho = 0$ at three range positions $vt = {v_{en}}t = 0,\,\;10,\textrm{ and 20}\textrm{.}$ Dimensionless parameter values: $a{}_1 = 1,\;\;{x_0} = 1$ and normalized values $c = 1,\;v = 4,\;{\beta _0} = 5,\;{\beta _1} = 2,\textrm{ and }{\bar{\beta }_2} = {10^{ - 1}}.$

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 figure: Fig. 5.

Fig. 5. (a) Normal dispersion: $|{{\textrm{Re}} \{{\varphi ({\rho ,\varsigma } )} \}} |\textrm{ }vs.\textrm{ }\varsigma \textrm{ and }\rho$; (b) anomalous dispersion: $|{{\textrm{Re}} \{{\varphi ({\rho ,\varsigma } )} \}} |\textrm{ }vs.\textrm{ }\varsigma \textrm{ and }\rho$; parameter values: ${a_1} = {10^{ - 1}}$ and normalized values ${\beta _0} = 5,{\bar{\beta }_2} = 1$ and $v = 2$.

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4. Conditional Galilei invariance of the bidispersive equation

An interesting question is whether the bidispersive equation can be invariant under an ordinary Galilean transformation. Let $T = \tau $ and $Z = z - v\tau .$ With these changes of coordinates, the bidispersive equation (14) changes to

$$\left( {i\frac{\partial }{{\partial Z}} \mp \frac{1}{2}{{\bar{\beta }}_2}\frac{{{\partial^2}}}{{\partial {T^2}}} + \frac{1}{{2{\beta_0}}}\nabla_ \bot^2} \right)\varphi (\rho ,\phi ,T,Z) ={\pm} \frac{1}{2}{\bar{\beta }_2}\left( {{v^2}\frac{{{\partial^2}}}{{\partial {Z^2}}} - 2v\frac{{{\partial^2}}}{{\partial Z\partial T}}} \right)\varphi (\rho ,\phi ,T,Z).$$

It follows, then, that for ordinary Galilei invariance, the right-hand side of Eq. (32) must vanish. It turns out that this is the case for any function $\varphi ({\rho ,\phi ,Z + vT/2} ).$ We consider, then, the left-hand side of Eq. (35) which can be written as

$$\left( {i\frac{\partial }{{\partial \varsigma }} \mp \frac{1}{2}{{\bar{\beta }}_2}\frac{{{v^2}}}{4}\frac{{{\partial^2}}}{{\partial {\varsigma^2}}} + \frac{1}{{2{\beta_0}}}\nabla_ \bot^2} \right)\varphi (\rho ,\phi ,\varsigma ) = 0;\;\varsigma = Z + \frac{v}{2}T = z - \frac{v}{2}\tau$$
indicating a decrease in dimensionality (descent) from $z,\tau ,\rho $ to $\varsigma ,\rho .$ It is important to compare the Galilean transformation for the Schrodinger and bidispersive equations. In the former, the transformed solution is characterized by an envelope translated as $z - v\tau $ and a phase factor containing $z - v\tau /2.$ No phase factor appears in the transformed bidispersive solution. Simply, a Galilean-invariant solution to the bidispersive equation is translated as $\varsigma = z - v\tau /2.$ This results in an effective envelope speed ${v_{en}} = v\,{v_g}/({v + 2{v_g}} ).$

4.1 Illustrative example: normal dispersion

We present, next, a specific transformed solution to the bidispersive equation:

$$\begin{array}{l} \varphi ({\rho ,\varsigma } )= \frac{1}{{Q({\rho ,\varsigma } )}}\exp [{\kern 1pt} ({{\beta_0}/B} )({i\varsigma - Q({\rho ,\varsigma } )} )]\,;\\ Q({\rho ,\varsigma } )= \sqrt {B{\rho ^2} + {{({{a_1} + i\varsigma } )}^2}} ;\;B = (1/4)({{\beta_0}{{\bar{\beta }}_2}{v^2}} ). \end{array}$$

This represents an infinite-energy X wave [14–18]. Both the envelope and phase speed are equal to $v\,{v_g}/({v + 2{v_g}} )$.

4.2 Illustrative example: anomalous dispersion

In the case of anomalous dispersion, Eq. (36) becomes

$$\left( {i\frac{\partial }{{\partial \varsigma }} + \frac{1}{2}{{\bar{\beta }}_2}\frac{{{v^2}}}{4}\frac{{{\partial^2}}}{{\partial {\varsigma^2}}} + \frac{1}{{2{\beta_0}}}\nabla_ \bot^2} \right)\varphi (\rho ,\phi ,\varsigma ) = 0;\;\varsigma = Z + \frac{v}{2}T.$$

A specific solution to this equation is given by

$$\varphi ({\rho ,z,t} )= {e^{ - i({{\beta_0}/B} )\varsigma }}\frac{{\sin \left[ {({{\beta_0}/B} )\sqrt {B{\rho^2} + {\varsigma^2}} } \right]}}{{\sqrt {B{\rho ^2} + {\varsigma ^2}} }};\;B = ({{\beta_0}{{\bar{\beta }}_2}{v^2}} )/4,$$
which is an infinite-energy MacKinnon-like structure [19,20]. Again, both the envelope and phase speeds are equal to the value $v\,{v_g}/({v + 2{v_g}} ),$ which is always subluminal provided that both v and ${v_g}$ are positive.

The infinite-energy X wave given in Eq. (37) and the infinite-energy MacKinnon -like structure in Eq. (39) are depicted below in Figs. 5(a) and 5(b), respectively.

5. Concluding remarks

The bidispersive equation, accounting for second-order temporal dispersive effects in a homogeneous lossless medium, has been examined in detail, first with respect to a Galilei transformation dual to that used in connection to the quantum mechanical Schrödinger equation. Furthermore, two Ansätzen have been introduced that allow the derivation of large classes of solutions of the bidispersive equation. Specific illustrative examples are given for both normal and anomalous dispersion. Finally, an investigation of a conditional ordinary Galilei transformation has revealed that it results in a decrease in dimensionality.

A note on superluminality underlying the dual Galilean transformation is appropriate. The scalar wave equation and, more generally, Maxwell’s equations in free space are invariant under the superluminal Lorentz transformations given in Eq. (8). The presence of a superluminal speed in finite-energy solutions to these equations does not contradict relativity. If parameters entering the solutions are chosen appropriately, a pulse moves superluminally with almost no distortion up to a certain distance, say ${z_d} = vt,$ and then it slows down to a luminal speed c, with significant accompanying distortion. Although the peak of the pulse does move superluminally up to ${z_d},$ it is not causally related at two distinct ranges ${z_1},{z_2} \in [{0,{z_d}} )$. Thus, no information can be transferred superluminally from ${z_1}$ to ${z_2}$.

The discussion in this article has been confined mostly to the implications of the dual Galilean transformations to a bidispersive equation. However, as in the case of the quantum mechanical Schrödinger equation whose solutions are invariant under other types of symmetry transformations, e.g., a conformal transformation, a dual such transformation applies to the bidispersive equation. Specifically, given a solution ${\varphi _ \mp }({x,y,\iota ,z} )$ of the bidispersive Eq. (14) for normal and anomalous dispersion, respectively, the conformal transformation

$$\begin{array}{l} {\psi _ \mp } = \frac{1}{{{{({1 + az} )}^{3/2}}}}\exp \left[ {i\frac{a}{{2({1 + az} )}}({ \mp {\tau^2}/{{\bar{\beta }}_2} + {\beta_0}{x^2} + {\beta_0}{y^2}} )} \right]\\ \textrm{ } \times {\varphi _ \mp }\left( {\frac{x}{{1 + az}},\frac{y}{{1 + az}},\frac{\tau }{{1 + az}},\frac{z}{{1 + az}}} \right), \end{array}$$
with a a positive parameter with units of inverse length, is also a solution.

Although the bidispersive equation is not form invariant under a dual Galilei transformation, a specific solution to the bidispersive equation remains a solution under such transformation. The same applies, for example, to the energy conservation law. Given the dimensionless bidispersive Eq. (22), the energy conservation laws for anomalous and normal dispersion are given as flows:

$$\begin{array}{l} P({X,Y,Z,T} )= {\varphi ^ \ast }({X,Y,Z,T} )\varphi ({X,Y,Z,T} ),\\ {{\vec{J}}_a}({X,Y,Z,T} )= \frac{1}{{2i}}({{\varphi^ \ast }{\nabla_{X,Y,T}}\varphi - \varphi {\nabla_{X,Y,T}}{\varphi^ \ast }} ),\\ {{\vec{J}}_n} = ({X,Y,Z,T} )= \frac{1}{{2i}}\left[ {{\varphi^ \ast }{\nabla_{X,Y}}\varphi - \varphi {\nabla_{X,Y}}\varphi - \left( {{\varphi^ \ast }\frac{{\partial \varphi }}{{\partial T}} - \varphi \frac{{\partial {\varphi^ \ast }}}{{\partial T}}} \right)} \right];\\ {\nabla _{X,Y,T}} \cdot {{\vec{J}}_{a,n}} + \frac{{\partial P}}{{\partial Z}} = 0. \end{array}$$

They are applicable for both the solutions of the bidispersive equation and the transformed ones.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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4. M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58(1), 1086–1093 (1998). [CrossRef]  

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10. J. M. Lévy-Leblond, “Une nouvelle limite non-relativiste du groupe de Poincaré,” Ann. Inst. Henri Poincaré, Sect. A 3(1), 1–12 (1965).

11. N. S. Sen Gupta, “On an analogue of the Galilei Group,” Il Nuovo Cim. XLIV, 512–517 (1966).

12. A. Chong W, H. Renninger, D. N. Christodoulides, et al., “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010). [CrossRef]  

13. N. K. Efremidis, Z. Chen, M. Segev, et al., “Airy beams and accelerating waves: An overview of recent advances,” Optica 6,686 (2019). [CrossRef]  

14. J. Y. Lu and J. F. Greenleaf, “Nondiffracting X Waves - Exact Solutions to Free Space Scalar Wave Equation and their Finite Aperture Realization,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 39(1), 19–31 (1992). [CrossRef]  

15. P. Saari and K. Reivelt, “Evidence of X-Shaped Propagation-Invariant Localized Light Waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997). [CrossRef]  

16. I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, et al., “Two Fundamental representations of localized wave pulse solutions to the scalar wave equation,” Progress in Electromagn. Res. PIER 19, 1–48 (1998). [CrossRef]  

17. J. Salo, J. Fagerholm, A. T. Freiberg, et al., “Unified description of nondiffracting X and Y waves,” Phys. Rev. E 62(3), 4261–4275 (2000). [CrossRef]  

18. P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69(3), 036612 (2004). [CrossRef]  

19. H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, Localized Waves (Wiley-Interscience, 2008).

20. H. E. Hernandez-Figueroa, E. Recami, and M. Zamboni-Rached, Non-Diffracting Waves (Wiley-VCH, 2014).

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (5)
Fig. 1.
Fig. 1. (a) $|{\varphi ({0,z,\tau } )} |\textrm{ }vs.\textrm{ }\tau \textrm{ and }z;$ with $a = 5 \times {10^{ - 2}}, {a_1} = 10,$ normalized values ${\beta _0} = 2, {\bar{\beta }_2} = 1$ and $v ={-} 1/2,$ (b) $|{\psi ({0,z,\tau } )} |\textrm{ }vs.\textrm{ }\tau \textrm{ and }z,$ with $a = 5 \times {10^{ - 2}},\;{a_1} = 10,\;{\beta _0} = 2,\;{\bar{\beta }_2} = {10^{ - 1}},\;v ={-} 2$.

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Fig. 2.
Fig. 2. (a) Surface plot of $|{{\textrm{Re}} \{{\psi (\rho ,T,z} \}} |\textrm{ vs}\textrm{. }T \textrm{ and }\rho $ at three range positions $({z = \,\, - 1/8,\;0,\textrm{ and }1/8} )$; (b) ${\textrm{Re}} \{{\psi (0,T,z} \}\textrm{ vs}\textrm{. }T \textrm{ }$ at three range positions $({z = \,\,0,\;1/4,\textrm{ and }1/2} )$. Parameter values: $a = {10^{ - 1}}$ and normalized values ${\beta _0} = 5,\bar{\beta } = 5 \times {10^{ - 2}}$ and $v = 1.2.$

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Fig. 3.
Fig. 3. (a) Surface plot of $|{\psi (\rho ,\varsigma } |\textrm{ vs}\textrm{. }\varsigma = z - {v_{en}}\textrm{t and }\rho $ at three range positions $vt = {v_{en}}t ={-} 30,\,\;0,\textrm{and 30}$ and (b) ${\textrm{Re}} \{{\psi ({\rho ,\varsigma } )} \}\textrm{ vs}\textrm{. }\varsigma $ for $\rho = 0$ at three range positions $ vt = {v_{en}}t = 0,\,\;30,\textrm{ and 50}\textrm{.}$ Dimensionless parameter values: $a{}_1 = 1,\;\;{x_0} = 1$ and normalized values $c = 1,\;v = 4,\;{\beta _0} = 5,\;{\beta _1} = 2,\textrm{ and }{\bar{\beta }_2} = {10^{ - 1}}.$

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Fig. 4.
Fig. 4. (a) Surface plot of $\left| {{\textrm{Re}} \{{\psi ({\rho ,\varsigma } )} \}} \right|\textrm{ vs}\textrm{. }\varsigma = z - {v_{en}}\textrm{t and }\rho $ at three range positions $vt = {v_{en}}t = - 10,\,\;0,\textrm{and 10}$ and (b) ${\textrm{Re}} \{{\psi ({\rho ,\varsigma } )} \}\textrm{ vs}\textrm{. }\varsigma $ for $\rho = 0$ at three range positions $vt = {v_{en}}t = 0,\,\;10,\textrm{ and 20}\textrm{.}$ Dimensionless parameter values: $a{}_1 = 1,\;\;{x_0} = 1$ and normalized values $c = 1,\;v = 4,\;{\beta _0} = 5,\;{\beta _1} = 2,\textrm{ and }{\bar{\beta }_2} = {10^{ - 1}}.$

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Fig. 5.
Fig. 5. (a) Normal dispersion: $|{{\textrm{Re}} \{{\varphi ({\rho ,\varsigma } )} \}} |\textrm{ }vs.\textrm{ }\varsigma \textrm{ and }\rho$; (b) anomalous dispersion: $|{{\textrm{Re}} \{{\varphi ({\rho ,\varsigma } )} \}} |\textrm{ }vs.\textrm{ }\varsigma \textrm{ and }\rho$; parameter values: ${a_1} = {10^{ - 1}}$ and normalized values ${\beta _0} = 5,{\bar{\beta }_2} = 1$ and $v = 2$.

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Equations (41)

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z ′ = γ s u b ( z − v t ) , x ′ = x , y ′ = y , t ′ = γ s u b ( t − z v / c 2 ) ; γ s u b = 1 / 1 − ( v / c ) 2 ; v < c
i h ∂ ∂ t φ ( x , y , z , t ) + h 2 2 m ( ∇ t 2 + ∂ 2 ∂ z 2 ) φ ( x , y , z , t ) = 0.
i h ( − v ∂ ∂ z ′ + ∂ ∂ t ′ ) φ ( x , y , z ′ , t ′ ) + h 2 2 m ( ∇ t 2 + ∂ 2 ∂ z ′ 2 ) φ ( x , y , z ′ , t ′ ) = 0.
ψ ( x , y , z , t ) = exp ⁡ [ i m v h ( z − 1 2 v t ) ] φ ( x , y , z − v t , t )
φ ( ρ , z , t ) = π 4 ( 2 m ℏ t 0 + i t ) 3 / 2 exp ⁡ [ − m 2 ℏ ( ρ 2 + z 2 t 0 + i t ) ] .
ψ ( ρ , z , t ) = exp ⁡ [ i h m v ( z − 1 2 v t ) ] π 4 ( 2 m ℏ t 0 + i t ) 3 / 2 exp ⁡ [ − m 2 ℏ ( ρ 2 + ( z − v t ) 2 t 0 + i t ) ] .
i k ∂ ∂ z φ ( x , z ) + 1 2 ∂ 2 ∂ x 2 φ ( x , z , ) = 0
z ′ = γ sup ( z − t c 2 / v ) , x ′ = x , y ′ = y , t ′ = γ sup ( t − z / v ) ; γ sup = 1 / 1 − ( c / v ) 2
∇ 2 u ( r → , t ) + β o p 2 ( − i ∂ / ∂ t ) u ( r → , t ) = 0.
F { β o p 2 ( − i ∂ / ∂ t ) u ( r → , t ) } = β 2 ( ω ) u ~ ( r → , ω )
u ( r → , t ) = φ ( r → , t ) e x p [ − i ω 0 ( t − z / v p h ) ] + c c , z ≥ 0.
[ ∇ ⊥ 2 + ∂ 2 ∂ z 2 + 2 i β c ( ω 0 ) ∂ ∂ z − β 2 ( ω 0 ) + ∑ m = 0 ∞ 1 m ! ∂ m ∂ ω m β 2 ( ω ) | ω = ω 0 ( i ∂ ∂ t ) m ] φ ( r → , t ) = 0.
[ ∇ ⊥ 2 + ∂ 2 ∂ z 2 + 1 v g r 2 ∂ 2 ∂ τ 2 − 2 1 v g r ∂ 2 ∂ z ∂ τ + 2 i β ( ω 0 ) ( ∂ ∂ z − 1 v g r ∂ ∂ τ ) − β 2 ( ω 0 ) ] φ ( r → , τ ) + ∑ m = 0 ∞ 1 m ! ∂ m ∂ ω m β 2 ( ω ) | ω = ω 0 ( i ∂ ∂ τ ) m φ ( r → , τ ) = 0.
( i ∂ ∂ z ∓ 1 2 β ¯ 2 ∂ 2 ∂ τ 2 + 1 2 β 0 ∇ ⊥ 2 ) φ ∓ ( x , y , τ , z ) = 0 ,
ψ ∓ ( x , y , τ , z ) = exp ⁡ [ ∓ i 1 v β ¯ 2 ( τ − z 2 v ) ] φ ∓ ( x , y , τ − z v , z )
( i ∂ ∂ z − 1 2 β ¯ 2 ∂ 2 ∂ τ 2 + 1 2 β 0 ∇ t 2 ) φ ( r → , τ ) = 0
φ ( ρ , z , τ ) = exp ⁡ [ − 1 12 ( 2 a − i z ) ( 2 a 2 − 4 i a z + z 2 − 6 τ β ¯ 2 ) ] A i ( τ β ¯ 2 − i a z − z 2 4 )   × 1 a 1 + i z exp ⁡ [ − β 0 2 ρ 2 a 1 + i z ] .
ψ ( ρ , τ , z ) = exp ⁡ [ − i 1 v β ¯ 2 ( τ − z 2 v ) ] φ ( ρ , τ − z v , z )
φ ( ρ , τ , z ) = 1 ( a + i z ) 3 / 2 exp ⁡ [ − 1 2 β 0 ρ 2 + τ 2 / β ¯ 2 a + i z ] , a > 0 ,
ψ ( ρ , τ , z ) = 1 ( a + i z ) 3 / 2 exp ⁡ [ i 1 v β ¯ 2 ( T + z 2 v ) ] exp ⁡ [ − 1 2 β 0 ρ 2 + T 2 / β ¯ 2 a + i z ]
Z = z / ( β 0 x o 2 ) , T = τ / τ 0 , X = x / x 0 , Y = y / x 0 .
( i ∂ ∂ Z ∓ 1 2 ∂ 2 ∂ T 2 + 1 2 ∇ ⊥ 2 ) φ ( X , Y , T , Z ) = 0.
( ∇ t 2 − ∂ 2 ∂ W 2 − λ 2 ) s ( X , Y , W ; λ ) ; λ = ( 2 + a 2 + 2 b ) 1 / 2 .
φ ( X , Y , T , Z ) = e i b Z e − i a T e i ( 1 + a 2 ) Z s [ X , Y , T − a Z ; ( 2 + a 2 + 2 b ) 1 / 2 ]
φ ( X , Y , T , Z ) = exp ⁡ ( i q Z ) Φ ( X , Y , T ; 2 q )
φ ( X , Y , T , Z ) = e i q Z e − i a ( T − a Z / 2 ) Φ ( X , Y , T − a Z ; 2 q ) .
s ( R , W ; λ ) = exp ⁡ ( − λ R 2 + ( a 1 + i W ) 2 ) / R 2 + ( a 1 + i W ) 2 ,
φ ( R , T , Z ) = e i b Z e − i a T e i ( 1 + a 2 ) Z exp ⁡ [ − ( 2 + a 2 + 2 b ) 1 / 2 R 2 + ( a 1 + i ( T − a Z ) ) 2 ] R 2 + ( a 1 + i ( T − a Z ) ) 2 .
φ ( ρ , z , t ) = e i A ( z − v p h t ) C B e r f c ( C B 2 ) ; A = v + v g 2 v 2 v g β ¯ 2 , C = ( a 1 − i z β 0 x 0 2 ) − 1 / 2 ; B = ρ 2 x 0 2 + [ a 1 − i 1 x 0 β 0 β ¯ 2 ( v + v g v v g ) ( z − v e n t ) ] 2 ;   v e n = v v g v + v g ,  v p h = 2 v v g 2 v + v g ; ρ = x 2 + y 2 .
( ∇ t 2 + ∂ 2 ∂ W 2 + λ 2 ) s ( X , Y , W ; λ ) ; λ = ( 2 + a 2 + 2 b ) 1 / 2 .
φ ( X , Y , T , Z ) = e − i b Z e i a T e − i ( 1 + a 2 ) Z s [ X , Y , T − a Z ; ( 2 + a 2 + 2 b ) 1 / 2 ]
s ( R , W ; λ ) = sin ⁡ ( λ R 2 + W 2 ) R 2 + W 2 .
φ ( R , T , Z ) = e − i b Z e i a T e − i ( 1 + a 2 ) Z sin ⁡ [ ( 2 + a 2 + 2 b ) 1 / 2 R 2 + ( T − a Z ) 2 ] R 2 + ( T − a Z ) 2 .
φ ( ρ , z , t ) = e i A ( z − v p h t ) C 3 e − C 2 B / 2 ; A = 2 v + v g 2 v 2 v g β ¯ 2 , C = ( a 1 + i z β 0 x 0 2 ) − 1 / 2 ; B = ρ 2 x 0 2 + [ 1 x 0 β 0 β ¯ 2 ( v + v g v v g ) ( z − v e n t ) ] 2 ;   v e n = v v g v + v g ,  v p h = 2 v v g 2 v + v g ; ρ = x 2 + y 2 .
( i ∂ ∂ Z ∓ 1 2 β ¯ 2 ∂ 2 ∂ T 2 + 1 2 β 0 ∇ ⊥ 2 ) φ ( ρ , ϕ , T , Z ) = ± 1 2 β ¯ 2 ( v 2 ∂ 2 ∂ Z 2 − 2 v ∂ 2 ∂ Z ∂ T ) φ ( ρ , ϕ , T , Z ) .
( i ∂ ∂ ς ∓ 1 2 β ¯ 2 v 2 4 ∂ 2 ∂ ς 2 + 1 2 β 0 ∇ ⊥ 2 ) φ ( ρ , ϕ , ς ) = 0 ; ς = Z + v 2 T = z − v 2 τ
φ ( ρ , ς ) = 1 Q ( ρ , ς ) exp ⁡ [ ( β 0 / B ) ( i ς − Q ( ρ , ς ) ) ] ; Q ( ρ , ς ) = B ρ 2 + ( a 1 + i ς ) 2 ; B = ( 1 / 4 ) ( β 0 β ¯ 2 v 2 ) .
( i ∂ ∂ ς + 1 2 β ¯ 2 v 2 4 ∂ 2 ∂ ς 2 + 1 2 β 0 ∇ ⊥ 2 ) φ ( ρ , ϕ , ς ) = 0 ; ς = Z + v 2 T .
φ ( ρ , z , t ) = e − i ( β 0 / B ) ς sin ⁡ [ ( β 0 / B ) B ρ 2 + ς 2 ] B ρ 2 + ς 2 ; B = ( β 0 β ¯ 2 v 2 ) / 4 ,
ψ ∓ = 1 ( 1 + a z ) 3 / 2 exp ⁡ [ i a 2 ( 1 + a z ) ( ∓ τ 2 / β ¯ 2 + β 0 x 2 + β 0 y 2 ) ]   × φ ∓ ( x 1 + a z , y 1 + a z , τ 1 + a z , z 1 + a z ) ,
P ( X , Y , Z , T ) = φ ∗ ( X , Y , Z , T ) φ ( X , Y , Z , T ) , J → a ( X , Y , Z , T ) = 1 2 i ( φ ∗ ∇ X , Y , T φ − φ ∇ X , Y , T φ ∗ ) , J → n = ( X , Y , Z , T ) = 1 2 i [ φ ∗ ∇ X , Y φ − φ ∇ X , Y φ − ( φ ∗ ∂ φ ∂ T − φ ∂ φ ∗ ∂ T ) ] ; ∇ X , Y , T ⋅ J → a , n + ∂ P ∂ Z = 0.
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